| L(s) = 1 | + (2.78 + 0.491i)2-s + (7.42 − 8.84i)3-s + (7.51 + 2.73i)4-s + (−0.626 + 0.228i)5-s + (25.0 − 20.9i)6-s + (−12.0 − 20.8i)7-s + (19.5 + 11.3i)8-s + (−9.10 − 51.6i)9-s + (−1.85 + 0.327i)10-s + (−12.1 + 21.0i)11-s + (80.0 − 46.2i)12-s + (114. + 136. i)13-s + (−23.3 − 64.1i)14-s + (−2.63 + 7.23i)15-s + (49.0 + 41.1i)16-s + (−51.6 + 293. i)17-s + ⋯ |
| L(s) = 1 | + (0.696 + 0.122i)2-s + (0.824 − 0.983i)3-s + (0.469 + 0.171i)4-s + (−0.0250 + 0.00912i)5-s + (0.695 − 0.583i)6-s + (−0.246 − 0.426i)7-s + (0.306 + 0.176i)8-s + (−0.112 − 0.637i)9-s + (−0.0185 + 0.00327i)10-s + (−0.100 + 0.174i)11-s + (0.555 − 0.320i)12-s + (0.676 + 0.806i)13-s + (−0.119 − 0.327i)14-s + (−0.0117 + 0.0321i)15-s + (0.191 + 0.160i)16-s + (−0.178 + 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 + 0.544i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.839 + 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.43957 - 0.721764i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.43957 - 0.721764i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.78 - 0.491i)T \) |
| 19 | \( 1 + (307. + 189. i)T \) |
| good | 3 | \( 1 + (-7.42 + 8.84i)T + (-14.0 - 79.7i)T^{2} \) |
| 5 | \( 1 + (0.626 - 0.228i)T + (478. - 401. i)T^{2} \) |
| 7 | \( 1 + (12.0 + 20.8i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (12.1 - 21.0i)T + (-7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-114. - 136. i)T + (-4.95e3 + 2.81e4i)T^{2} \) |
| 17 | \( 1 + (51.6 - 293. i)T + (-7.84e4 - 2.85e4i)T^{2} \) |
| 23 | \( 1 + (476. + 173. i)T + (2.14e5 + 1.79e5i)T^{2} \) |
| 29 | \( 1 + (-346. + 61.1i)T + (6.64e5 - 2.41e5i)T^{2} \) |
| 31 | \( 1 + (175. - 101. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + 1.10e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-226. + 270. i)T + (-4.90e5 - 2.78e6i)T^{2} \) |
| 43 | \( 1 + (-2.77e3 + 1.01e3i)T + (2.61e6 - 2.19e6i)T^{2} \) |
| 47 | \( 1 + (357. + 2.02e3i)T + (-4.58e6 + 1.66e6i)T^{2} \) |
| 53 | \( 1 + (552. - 1.51e3i)T + (-6.04e6 - 5.07e6i)T^{2} \) |
| 59 | \( 1 + (-4.57e3 - 806. i)T + (1.13e7 + 4.14e6i)T^{2} \) |
| 61 | \( 1 + (6.93e3 + 2.52e3i)T + (1.06e7 + 8.89e6i)T^{2} \) |
| 67 | \( 1 + (5.19e3 - 915. i)T + (1.89e7 - 6.89e6i)T^{2} \) |
| 71 | \( 1 + (-394. - 1.08e3i)T + (-1.94e7 + 1.63e7i)T^{2} \) |
| 73 | \( 1 + (3.96e3 + 3.32e3i)T + (4.93e6 + 2.79e7i)T^{2} \) |
| 79 | \( 1 + (-6.58e3 + 7.84e3i)T + (-6.76e6 - 3.83e7i)T^{2} \) |
| 83 | \( 1 + (-3.37e3 - 5.85e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-3.75e3 - 4.47e3i)T + (-1.08e7 + 6.17e7i)T^{2} \) |
| 97 | \( 1 + (1.23e4 + 2.17e3i)T + (8.31e7 + 3.02e7i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.12745167868701493437621502702, −13.97251691561222525205328101905, −13.29468577341370268593115382317, −12.28215782420139287304300790136, −10.71543557142723179168238059359, −8.756887143617395663749857412462, −7.47509418023861155007573435652, −6.29187123564477528772202434841, −3.97066990847601167529958434900, −2.02671325248967337273781661150,
2.87226920850735682587137379962, 4.25035723390363498311601092164, 5.98335804655266178598082814669, 8.136607078516541506079414828227, 9.504228946396949066690861492820, 10.66223850404284089743210445919, 12.16272236274095105260199873994, 13.52331400323613999510424280688, 14.52797286378168546779016415176, 15.59315741099222564024536837779
